Solution 5.8.1.29 G H C R + - Figure 1: For the system shown above GH(s)= K(s +4) s(s + 1)(s +20) The rst step is to plot the poles and zeros of GH in the s-plane and then nd the root locus on the real axis. The shaded regions of the real axis in Figure 2 shows where the root locus occurs. The rule is that root locus occurs on the real axis to the left of an odd countofpoles and zeros. That is, if you stand on the real axis and look to your right you must countan odd numberofpoles and zeros. The next step is to compute the asymptotes. The numberofasymptotes is p ex =Number of poles of GH ;Numberof nite zeros of GH: Im(s) Re(s) -1 -20 -4 Figure 2: 1 The asymptotes occur at the angles  ` =  1+2` p ex  180  ` =0;;1;;:::p ex ;1: In the presentcase there are three poles and one nite zero so p ex =2. Thus, there are two asymptotes at  0 =  1+(2)(0) 2  180  =90   1 =  1+(2)(1) 2  180  =270  The asymptotes intersect the real axis at  i = Sum of poles of GH ;Sum of zeros of GH p ex = [(0) + (;1) + (;20)];[;4] 2 = ;8:5 Since there is one nite zeros and three poles, twoclosed loop poles will migrate to zeros at in nity,onesuchzeroatthe`end' of eachasymptote. The root locus is shown in Figure 3 2 Im(s) Re( -1 -20 -4 Figure 3: Completed Root Locus 3